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\centerline{\bf Algebraic Topology}
\centerline{Homework 8: Due Wednesday, October 20}      % e.g. September 13
\centerline{\bf Revised October 18}
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{\bf Problem 1} Back in early September, we proved that the fundamental
group of a circle was infinite cyclic using Lebesgue numbers and an 
argument about angles. I want you to take
that ``keep track of angles'' idea to its logical conclusion.  
Use the properties of the cover $R \to S^1$,
$x \to e^{ix}$ and Lemmas 3.1, 3.2 and 3.3 to prove that $\pi_1(S^1)=Z$. 
Do {\it not} use Lebesgue numbers or partitions of the interval $[0,1]$
into pieces (except insofar as these concepts were required to prove
Lemmas 3.1-3.3)

{\bf Problem 2}. Let $X$ be a path-connected and locally path-connected
Hausdorff space, and consider the category $C_1$ 
whose objects are covering spaces
of $X$, and whose morphisms are homomorphisms of covering spaces (as defined
on page 130). Show that $C_1$ need not have a universal object. [Hint: it's
sufficient to consider $X=S^1$.]

{\bf Problem 3.}  Now pick a base point $x \in X$, and consider the
category $C_2$ whose objects are {\it based} covering spaces $(\tilde
X, \tilde x,p)$ where $p(\tilde x)=x$, and whose morphisms are
required to take base points to base points. Show that $(\tilde X,
\tilde x, p)$ is a universal object if $\tilde X$ is
simply-connected. (The converse is also true but is harder.)

{\bf Problem 4} Page 132, problem 6.4

The following two problems were originally assigned, but are {\bf no longer 
due on October 20}.  They will be included in homework \#8, due on October
27.

{\bf Problem 5}
Page 135, problem 7.2 -- For example 2.4, you can restrict attention to the
cover $\R^2 \to T^2$. 

{\bf Problem 6} Page 144, problem 10.1  

\bye
